Fixed points and controllability in delay systems

FIXED POINTS AND CONTROLLABILITY IN DELAY SYSTEMS

Received 9 December 2004; Revised 28 June 2005; Accepted 6 July 2005

Schaefer’s fixed point theorem is used to study the controllability in an infinite delay sys-temx(t)=G(t,xt) + (Bu)(t). A compact map or homotopy is constructed enabling us to show that if there is ana prioribound on all possible solutions of the companion control systemx(t)=λ[G(t,xt) + (Bu)(t)], 0< λ <1, then there exists a solution forλ=1. The a prioribound is established by means of a Liapunov functional or applying an integral inequality. Applications to integral control systems are given to illustrate the approach.

Copyright © 2006 H. Gao and B. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

This paper is concerned with the problem of controllability in an infinity delay system

x(t)=Gt,xt

+ (Bu)(t), t∈J=[0,b], (1.1)

wherex(t)∈Rn_,x

t(θ)=x(t+θ) for −∞< θ≤0,u(t), the control, is a real m-vector valued function onJ,B is a bounded linear operator acting onu, andGis defined on J×CwithCbeing the Banach space of bounded continuous functionsφ: (−∞, 0]→Rn with the supremum norm| · |C.

The problem of controllability in delay systems has been the subject of extensive inves-tigations by many scientists and researchers for over half of a century. A large number of applications have appeared in biology, medicine, economics, engineering, and informa-tion technology. Many actual systems have the property of “after-effect,” that is, the future state depends not only on the present, but also on the past history. It is well-known that such a delay factor, when properly controlled, can essentially improve system’s qualitative and quantitative characteristics in many aspects. For historical background and discus-sion of applications, we refer the reader to the work of Balachandran and Sakthivel [1], Chukwu [4], G ´orniewicz and Nistri [8], and references therein.

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 41480, Pages1–14

Equation (1.1) describes the state of a system (physical, chemical, economic, etc.) whose evolution in timetis governed byG(t,xt) + (Bu)(t). In general, we view a solu-tion of (1.1) as a function ofu(t) so that the behavior of the system depends on (or is controlled by) the choice ofuwithin a setU given in advance. Assume that two states of the system are given, one to be considered as an initial stateφ, and the other as a final stateγ. The problem of controllability is to determine whether there are available controls which can transfer the statexfromφtoγalong a solution (1.1), that is, whether there exists au0∈U such that ˙x(t)=G(t,xt) +Bu0(t) has a solution joiningφandγ. When this is possible for arbitrary choice ofφandγ, we can say, roughly speaking, that system (1.1) is controllable by means ofU.

The main method of proving controllability has been to write (1.1) as an integral equa-tion

x(t)=x0+ _t

0

Gs,xs

+Bu(s)ds (1.2)

viewing the right-hand side as a mappingPxon an appropriate space, whenuis properly chosen in terms ofx. Then, apply a fixed point theorem, say Schauder’s, to the mapping PwhenP:K→Kis compact for a closed, convex subsetKof a Banach space. However, P, in general, does not satisfy this condition unless the growth ofG(t,xt) inxis restricted. This presents a significant challenge to investigators. A modern approach to such a prob-lem is to use topological degree or transversality method (see G ´orniewicz and Nistri [8]). Here we will use a fixed point theorem of Schaefer [12] which is a variant of the nonlinear alternative of Leray-Schauder degree theory, but much easier to use.

The paper is organized as follows. InSection 2, we prove our main result on controlla-bility of (1.1). Applications to specific systems are given inSection 3which also contains some general results and remarks concerning the approach.

2. Controllability

We start this section with some descriptions of spaces associated with our discussion. Let R=(−∞,∞),R−=(−∞, 0], andR+_{=[0,∞), respectively, and| · |denote the Euclidean} norm inRn_{. For ann×nmatrixA=(a}

i j)n×n, defineA =sup|x|≤1|Ax|. We denote by C(X,Y) the set of bounded continuous functionsφ:X→Y for normed spacesXandY. ForJ given in (1.1), we defineC0= {φ∈C(J,Rn_{) :φ(0)=0}with the supremum norm}

| · |C0, and setC0(μ)= {φ∈C0:|φ|C0≤μ}. LetL2(J) denote the Banach space of square Lebesgue integrable functionsx:J→Rn_{with the norm|x|}

2=(

J|x(s)|2ds)1/2with the understandingx=0∈L2_{(J) if and only ifx(t)=0 a.e. onJ. We denote byUthe space of} admissible controls and chooseUas a complete subspace ofL2_{(J) with a norm| · |}

U. The symbol · is reserved for the norm of a linear operator.

We assume thatB:U→L2_{(J) is a bounded linear operator. For a givenu∈U, we say} thatx: (−∞,b]→Rn_{is a solutionx=x(t,φ) of (1.1) onJwith initial functionφifxis} absolutely continuous onJand satisfies

x(t)=Gt,xt

+ (Bu)(t), a.e.t∈J=[0,b] (2.1)

Our result rests on a fixed point theorem of Schaefer [12]. Its relation to Leray-Schauder degree theorem is explained in Smart [13, page 29]. Schaefer’s theorem has been used in a variety of areas in differential equations and control theory (see Burton [2], Burton and Zhang [3], Balachandran and Sakthivel [1]).

Theorem2.1 (Schaefer). LetV be a normed space,F a continuous mapping ofV intoV which is compact on each bounded subset ofV. Then either

(i)the equationx=λF(x)has a solution forλ=1, or (ii)the set of all such solutionsx, for0< λ <1, is unbounded.

If we viewUλ(x)=λF(x) in Schaefer’s theorem as a homotopy, then it can be restated in the form of Leray-Schauder Principle (cf. Zeidler [14, page 245]), which is often used in application.

Definition 2.2. System (1.1) is said to be controllable on the intervalJ if for eachφ∈C and γ∈Rn_{, there exists a control u∈U such that the solutionx(t)=x(t,φ) of (1.1)} satisfiesx(b)=γ.

Throughout this paper, we letφ∈Candγ∈Rn_{be arbitrary, but fixed. For eachy∈} C0, we define

¯ y(t)=

⎧ ⎨ ⎩y

(t) +φ(0) t∈J,

φ(t) t∈R−. (2.2)

This implies that ¯y0=φ; that is, ¯y(s)=φ(s) for alls∈R−. For each continuous function z: (−∞,b]→Rwithz0=φ, we define

[z]=γ−φ(0)−

JG

s,zsds. (2.3)

We now introduce a companion to (1.1)

x(t)=λGt,xt

+ (Bu)(t), a.e.t∈J=[0,b] (1.1_λ)

forλ∈[0, 1] and make the following assumptions. (H1) The linear operatorT:U→Rndefined by

T(u)=

JBu(s)ds (2.4)

is invertible; that is, for each α∈Rn_{, there exists a unique u}

α ∈U such that α=_J(Buα)(s)ds.

(H2) For each y∈C0,G(s, ¯ys) is Lebesgue measurable insonJ, and for eachμ >0, there exists an integrable functionMμ:J→R+such that|G(s, ¯ys)| ≤Mμ(s) for alls∈J whenevery∈C0(μ).

(H3) For anyε >0 and y∈C0, there exists aδ >0 such that [x∈C0,|x−y|C0< δ], imply

J _G

s, ¯xs

(H4) For each φ∈C and γ∈Rn, there exists a constant L=L(φ,γ)>0 such that

|xλ(t)| ≤Lfor allt∈J wheneverxλ(t)=x(t,φ) is a solution of (1.1λ) withu(t)=uα(t) forα=[xλ] andλ∈(0, 1).

Theorem2.3. If (H1)–(H4) are satisfied, then (1.1) is controllable onJ. Proof. Letφ∈Candγ∈Rn_{be fixed. We define a functionF:C0→C0by}

(F y)(t)=

_t

0

Gs, ¯ys+Bu[ ¯y](s)

ds (2.6)

for each y∈C0 and t∈J. It is clear thatF is well-defined. For the linear operator T defined in (H1), we have

_T(u)=

JBu(s)ds ≤

J

_Bu(s)ds≤b1/2_|Bu|

2≤b1/2B|u|U. (2.7)

Thus, T :U →R is bounded, and hence T−1 _{is also bounded, say T}−1 _{≤M1 (see} Friedman [6, page 143]).

We now show thatF is continuous onC0. For eachε >0 and y∈C0, by (H3) there exists aδ >0 such that [x∈C0,|x−y|C0< δ] imply

J _G

s, ¯xs−Gs, ¯ysds <_{1 +} ε

b1/2_BT−1. (2.8)

Observe

J

_{Bu[ ¯x](s)−Bu[ ¯y](s)ds}

≤b1/2Bu[ ¯x]−Bu[ ¯y]2≤b1/2Bu[ ¯x]−u[ ¯y]_U=b1/2BT−1[ ¯x]−T−1[ ¯y]_U

≤b1/2BT−1[ ¯x]−[ ¯y]≤b1/2BT−1

J

Gs, ¯xs

−Gs, ¯ys

ds, (2.9)

where [ ¯x]=γ−φ(0)−_JG(s, ¯xs)dsand [ ¯y]=γ−φ(0)−

JG(s, ¯ys)ds. Thus, we obtain _{F(x)(t)−F(y)(t)≤}

J _G

s, ¯xs

−Gs, ¯ysds+

J _{Bu[ ¯}

x](s)−Bu[ ¯y](s)ds

≤1 +b1/2BT−1

J _G

s, ¯xs

−Gs, ¯ysds < ε.

(2.10)

This implies that|F(x)−F(y)|C0< εwhenever|x−y|C0< δ, and henceFis continuous onC0.

Next, we show that for eachμ >0, the set{F(y) :y∈C0(μ)}is uniformly bounded and equicontinuous. By the definition ofF, we have

_F(y)(t)=t 0G

s, ¯ys

ds+

_t

0

Bu[ ¯y]

(s)ds≤

JMμ(s)ds+b 1/2

Notice that [ ¯y]=_JBu[ ¯y](s)ds=T(u[ ¯y]) so thatu[ ¯y]=T−1[ ¯y]. Therefore,

_{u[ ¯}

y]_U≤T−1[ ¯y]≤M1

|γ|+φ(0)+

J _G

s, ¯ysds

≤M1

|γ|+φ(0)+

JMμ(s)ds

=:M2.

(2.12)

Combine (2.11) and (2.12) to obtain |F(y)|C0≤M3 for some M3>0 and for all y∈ C0(μ). Thus,{F(y) :y∈C0(μ)}is uniformly bounded. Now let t1,t2∈J witht1< t2. Then fory∈C0(μ), we have

_F(y)

t2−F(y)t1≤

_t2

t1

Mμ(s)ds+ _t2

t1

_{Bu[ ¯y](s)ds}

≤

_t2

t1Mμ(s)ds+

t2−t11/2 t2

t1

_{Bu[ ¯y](s)}2 ds

1/2

≤

_t2

t1Mμ(s)ds+

t2−t11/2Bu[ ¯y]_U

≤

_t2

t1Mμ(s)ds+

t2−t11/2BM2

(2.13)

by (2.12). Thus,{F(y) :y∈C0(μ)}is uniformly bounded and equicontinuous onJ, and henceFis compact by Ascoli-Arzela’s theorem.

Finally, suppose thatx=λF(x) for somex∈C0and 0< λ <1; that is,

x(t)=λ _t

0G

s, ¯xs

ds+ _t

0Bu[ ¯x](s)ds

(2.14)

fort∈J. Differentiate (2.14) with respect totto obtain

¯

x(t)=x(t)=λGt, ¯xt

+Bu(t), a.e.t∈J (2.15)

with u=u[ ¯x]. From (2.15), we see that ¯x is a solution of (1.1λ) with ¯x0=φ. More-over, |x¯|C0≤L by (H4). This implies that|x|C0≤L+|φ(0)|. Therefore, alternative (i) of Theorem 2.1must hold, and there exists y∈C0 such that y=F(y). Following the argument in (2.14) and (2.15), we see that ¯yis a solution of (1.1) with ¯y0=φand

¯

y(b)−y¯(0)=

_b

0G

s, ¯ysds+ _b

0Bu[ ¯y](s)ds= _b

0G

s, ¯ysds+ [ ¯y]

=

_b

0G

s, ¯ysds+γ−φ(0)− _b

0G

s, ¯ysds.

(2.16)

Since ¯y(0)=φ(0), we obtain ¯y(b)=γ, and so (1.1) is controllable on the intervalJwith

3. Examples and remarks

In this section, we give several examples to illustrate how to applyTheorem 2.3to some specific equations and systems. Our emphasis will be on obtaining a priori bounds. The examples are shown in simple forms for illustrative purposes, and they can easily be gen-eralized.

Example 3.1. Consider the control problem

x(t)=Ax(t) + _t

−∞E

t,s,x(s)ds+u(t), a.e.t∈J, (3.1)

whereA=(ai j)n×nis ann×nmatrix,E:Ω×Rn→Rnis measurable withΩ= {(t,s)∈ R2_{:t≥s}, andu(t) is an arbitrary control (to be determined later).}

It is well known that

e−At=ω1(t)I+ω2(t)A+···+ωn(t)An−1, (3.2)

whereωi:R→R(i=1, 2,...,n) are continuous and linearly independent (see Godunov [7, page 32]). Denote the space of linear span of functionsw1,w2,...,wnby

spanω1,ω2,...,ωn. (3.3)

Letc∈Rn_{be fixed. We defineUas}

U=u=u∗c|u∗∈spanω1,ω2,...,ωn

, (3.4)

and view (U,| · |U),| · |U= | · |2, as a complete subspace of (L2(J),| · |2). Introduce a transformationy(t)=e−At_{x(t) to write (3.1) as}

y(t)=Gt,yt

+e−Atu(t), a.e.t∈J, (3.5)

where

Gt,yt

=

_t

0e

−At_Et,s,eAs_y(s)ds+ 0

−∞e

−At_{Et,s,φ(s)ds. (3.6)}

We assume that 0

−∞E

t,s,φ(s)ds is integrable onJwith respect tot (3.7)

for eachφ∈C. It is clear that (3.1) is controllable if and only if (3.5) is controllable. We now write

(Bu)(t)=e−At_{u(t) (3.8)}

and show that the linear operatorT:U→Rn_{defined by}

T(u)=

J(Bu)(s)ds=

Je

is invertible. To this end, we assume

spanc,Ac,A2c,...,An−1c=Rn. (3.10)

Let α∈Rn_{. We will find a uniqueu∈U such thatT(u)=α. By (3.10), there exists a} uniquea=(a1,a2,...,an)T∈Rnsatisfying

α=a1c+a2Ac+···+anAn−1c. (3.11)

Sinceω1,ω2,...,ωnare linearly independent, we have

=: detωi,ωj_n×n

=0, (3.12)

whereωi,ωj =

Jwi(s)wj(s)dsis the inner product inL2(J). Thus, the system of equa-tions

ω1,ωjk1+ω2,ωjk2+···+ωn,ωjkn=aj (3.13)

(j=1, 2,...,n) has a unique solution (k1,k2,...,kn). We now define

u∗(t)=ω1(t)k1+ω2(t)k2+···+ωn(t)kn. (3.14)

Multiply (3.14) bye−As_{cand integrate onJto obtain}

Je

−As_u(s)ds=

Je

−As_cu∗_(s)ds= J

_n

j=1

ωj(s)Aj−1c

_n

i=1 ωi(s)ki

ds

=n

i=1

ωi,ω1

kic+···+ n

i=1

ωi,ωn

kiAn−1c

=a1c+a2Ac+a3A2_c+···+a

nAn−1c=α.

(3.15)

This implies thatTis invertible.

By Cramer’s rule, we writekj in (3.13) as kj= j(a)/ wherej(a) is the n×n determinant obtained by replacing thejth column ofbya=(a1,a2,...,an)Tdefined in (3.11). Thus

_u∗_(t)≤n j=1

_j(a)

|| wj(t). (3.16)

Since

whereCi j is the cofactor ofwi,wjin the matrix (wi,wj)n×n, and1|α| ≤ |a| ≤2|α| for some positive constants1and2, we have

_{j(a)≤ |a|}n i=1

_{Ci j}2

≤2|α|

n

i=1 _{Ci j}2

, (3.18)

_b

0

_e−At_{u(t)dt≤ |c|}n j=1

2

||

n

i=1 _C

i j 2b

0e At_w

j(t)dt|α| =:K|α|. (3.19)

This implies that

J _e−At_u

α(t)dt≤K|α|. (3.20)

Theorem3.2. Suppose that (3.7), (3.10), and the following conditions hold.

(i) There exist a positive constantM and a measurable functionq:Ω0→R+,Ω0=

{(t,s)∈Rn_{: 0≤s≤t≤b}such that}

_{E(t,s,x)≤q(t,s)}

|x|+M. (3.21)

(ii) For anyμ >0, there existskμ:Ω0→R+with

J _t

0kμ(t,s)dsdt <∞such that _{E(t,s,x)−E(t,s,y)≤k}

μ(t,s)|x−y| (3.22)

for allt,s∈Ω0,|x| ≤μ, and|y| ≤μ. (iii)

(K+ 1) _b

0 _t

0e A(t+s)

q(t,s)dsdt=:r <1, (3.23)

whereKis defined in (3.19). Then (3.1) is controllable.

Proof. It suffices to show that (3.5) is controllable; that is, for anyφ∈Candγ1∈Rn_{, there} exists a controlu∈Usuch that the solutiony(t)=y(t,φ) of (3.5) satisfiesy(b)=γ1. We have shown that (H1) holds. Forφ∈Candy∈C0(μ), it is clear thatG(t, ¯yt) is measurable int. Forφ∈Candy∈C0(μ), we also have

_G

t, ¯yt= t

0e

−At_Et,s,eAs_{y(s) +φ(0)ds+} 0

−∞e

−At_Et,s,φ(s)ds

≤

_t

0e

At_q(t,s)eAs_{y(s)+φ(0)+Mds+e}Ab 0

−∞E

t,s,φ(s)ds

≤μ+φ(0)+M t 0e

A(t+s)

q(t,s)ds+eAb 0

−∞E

t,s,φ(s)ds

=:Mμ(t).

Thus, (H2) holds. Next, letμ1>0 andx,y∈C0(μ1). By (ii), there existskμ(t,s),μ=(μ1+

|φ(0)|)eAb_{, such that E}

t,s,eAsx(s) +φ(0)−Et,s,eAsy(s) +φ(0)≤eAbkμ(t,s)x(s)−y(s) (3.25)

for all (t,s)∈Ω0. This yields _b

0 _G

t, ¯xt

−Gt, ¯ytdt

=

_b

0 t

0e

−At_Et,s,eAs_{x(s) +φ(0)ds− t}

0e

−At_Et,s,eAs_{y(s) +φ(0)dsdt}

≤e2Ab

J _t

0kμ(t,s)dsdt|x−y|C0

(3.26)

for allx,y∈C0(μ1), and hence (H3) is satisfied.

We now show that (H4) holds. Lety=y(t,φ) be a solution of

y(t)=λGt,yt

+e−Atu(t), a.e.t∈J (3.2λ)

withλ∈(0, 1),y0=φ, and

Je

−As_{u(s)ds=γ1−φ(0)−}

JG

s,ys

ds. (3.27)

Integrate (3.2λ) from 0 totto obtain

y(t)=φ(0) +λ _t

0G

τ,yτ

dτ+λ _t

0e

−Aτ_{u(τ)dτ. (3.28)}

Thus,

_{y(t)≤φ(0)+}b 0

_G

τ,yτds+ _b

0

_e−Aτ_{u(τ)dτ. (3.29)}

It follows from (3.20) that _b

0

_e−Aτ_{u(τ)dτ≤Kγ1+φ(0)+ b}

0 _G

τ,yτdτ

. (3.30)

Substituting (3.30) into (3.29), we arrive at

_{y(t)≤Kγ1+ (1 +K)φ(0)+ (1 +K)}b 0

_G

τ,yτdτ

≤Kγ1+ (1 +K)φ(0)+ (1 +K) _b

0

_−∞0 e−AτEτ,s,φ(s)dsdτ

+ (1 +K) _b

0 _τ

0

≤Kγ1+ (1 +K)φ(0)+ (1 +K) _b

0

_−∞0 e−Aτ_{Eτ,s,φ(s)dsdτ}

+ (1 +K) _b

0 _τ

0e

At_q(τ,s)eAs_y(s)+Mdsdτ

=: (1 +K) _b

0 _τ

0e

A(t+s)_{q(τ,s)y(s)dsdτ+M}∗

≤r|y|C0+M∗.

(3.31)

This yields|y|C0≤M∗/(1−r), and hence (H4) holds. ByTheorem 2.3, system (3.5) is

controllable. The proof is complete.

Remark 3.3. A more general expression can be introduced in the control term in (3.1) such asB(t)u(t) whereB(t) is ann×mmatrix function andu(t)∈Rm_{. WhenE(t,s,x)≡} 0, (3.1) is reduced to

x(t)=Ax+u∗(t)c. (3.32)

A classical result states that (3.32) is controllable if and only if (3.10) holds (see Godunov [7, page 211] and Conti [5, page 98]).

Example 3.4. Consider the scalar Volterra equation

x(t)= −

_t

−∞a(t−s)q

x(s)ds+u(t), a.e.t∈J, (3.33)

wherea:R→R,q:R×Rare continuous, andu∈U. For a fixedξ:J→R+_with|ξ| 2=1, we define

U=u∈L2_{(J) :u=kξ,k∈R (3.34)}

with|u|U= |u|2= |k||ξ|2= |k|. Therefore,Uis a Banach space (dimension 1) with| · |U. Observe that (3.33) can be written in the form of (1.1) with

Gt,xt= − _t

−∞a(t−s)q

x(s)ds (3.35)

andB:U→L2_{(J) being the identity operator (B =1). DefineT:U→Rby}

T(u)=

JBu(s)ds=

Ju(s)ds. (3.36)

Theorem3.5. Suppose the following conditions hold.

(˜i)_−∞0 a(t−s)q(φ(s))dsis integrable onJwith respect totfor eachφ∈C. (˜ii)a∈C(R,R) witha(t)≥0,a(t)≤0,a(t)≥0 for allt≥0.

( ˜iii) There are positive constantsαandKsuch thatQ(x) +K >0 and _{q(x)≤αQ(x) +K ∀x∈Rwith lim}

|x|→∞Q(x)= ∞, (3.37)

whereQ(x)=0xq(s)ds. ( ˜iv)

α2

2

J _t

0a(s)dsdt=:η <1. (3.38)

Then (3.33) is controllable.

Proof. We verify that conditions (H2)–(H4) hold. For eachφ∈Candx∈C0(μ), we have _G

t, ¯xt=

_−∞0 a(t−s)qφ(s)ds+ _t

0a(t−s)q

x(s) +φ(0)ds

≤

0

−∞a(t−s)q

φ(s)ds+ _t

0a(t−s) _q

x(s) +φ(0)ds

≤

0

−∞a(t−s)q

φ(s)ds+ _t

0a(s)ds qμ=:Mμ(t),

(3.39)

whereqμ=sup{|q(z)|:|z| ≤μ+|φ(0)|}. Thus, (H2) is satisfied.

Now letμ >0 andx,y∈C0(μ). Sinceqis uniformly continuous on [−(μ+|φ(0)|),μ+

|φ(0)|], for anyε >0, there exists aδ >0 such that|x−y|C0< δimplies _q

x(s) +φ(0)−qy(s) +φ(0)< ε (3.40)

for alls∈J, and so

J _G

s, ¯xs

−Gs, ¯ysds=

J t

0a(t−s)

qx(s) +φ(0)−qy(s) +φ(0)dsdt

≤

J _t

0a(s)dsdt ε≤ 2ε α2.

(3.41)

This shows (H3) holds.

Letx(t)=x(t,φ) be a solution of

x(t)=λ

−

_t

−∞a(t−s)q

x(s)ds+u(t)

, a.e.t∈J, (3.42)

whereλ∈(0, 1),x0=φ, and

Ju(s)ds=γ−φ(0) +

J _τ

−∞a(τ−s)q

We apply Liapunov’s direct method to derive a priori bounds onx. Define

E(t)=Qx(t)+K+1 2λa(t)

_t

0q

x(s)ds 2

−1

2λ _t

0a

_(t−s)t sq

x(ν)dν

2

ds (3.44)

for allt≥0 and setV(t)=E(t). Then

V(t)= 1

2E(t)

q(x)λ

−

_t

−∞a(t−s)q

x(s)ds+u(t)

+1 2λa

_(t)t 0q

x(s)ds

2 +λa(t)

_t

0q

x(s)ds

qx(t)

−1

2λ _t

0a

_(t−s)t sq

x(ν)dν

2 ds

−λ _t

0a

_(t−s)t sq

x(ν)dν

ds qx(t).

(3.45)

Integrating by parts in the last term, we get

−λ _t

0a

_(t−s)t sq

x(ν)dν

ds

=λa(t−s) _t

sq

x(ν)dν|t 0+λ

_t

0a(t−s)q

x(s)ds

= −λa(t) _t

0q

x(ν)dν+λ _t

0a(t−s)q

x(s)ds.

(3.46)

Substitute (3.46) into (3.45) and apply condition ( ˜iii) to obtain

V(t)= 1 2E(t)

λqx(t)u(t)−λqx(t) 0

−∞a(t−s)q

φ(s)ds

+1 2λa

_(t)t 0q

x(s)ds

2

−1

2λ _t

0a

_(t−s)t sq

x(ν)dν

2 ds

≤ 1

2Q(x) +K

qx(t)u(t)+ _∞

0 a(ν)dν_tsup_∈R− _q

φ(s)

≤α

2

u(t)+ _∞

0 a(ν)dν_ssup_∈R− _q

φ(s)

=:α

2u(t)+Γ1.

IntegratingV(t) from 0 totand using (3.43), we find

V(t)≤V(0) +α 2

_b

0

_{u(s)ds+bΓ1=Q}

x(0)+K+α 2 b

0u(s)ds +bΓ1

=Qφ(0)+K+bΓ1+α 2

γ−φ(0) +

J _τ

−∞a(τ−s)q

x(s)dsdτ

≤α

2

J _τ

0a(τ−s) _q

x(s)dsdτ+Qφ(0)+K+bΓ1

+α 2

|γ|+φ(0)+

J

_−∞0 a(τ−s)qφ(s)dsdt

=:α 2

J _τ

0a(τ−s) _q

x(s)dsdτ+Γ2.

(3.48)

Observe

Qx(t)+K≤V(t)≤α

2

J _τ

0 a(τ−s) _q

x(s)dsdτ+Γ2

≤

sup

s∈J

Qx(s)+K

α2 2

J _τ

0a(ν)dν+Γ2

≤ηsup s∈J

Qx(s)+K+Γ2.

(3.49)

This implies

sup s∈J

Qx(s)+K≤ Γ2

(1−η). (3.50)

SinceQ(x)→ ∞as|x| → ∞, there exists a positive constantL=L(Γ2,η) such that|x(t)|< Lfor allt∈J. Therefore, (H4) holds. ByTheorem 2.3, we conclude that (3.33) is

control-lable.

Remark 3.6. Equations such as (3.33) have been the center of much interest for a long time in connection with a problem of reactor dynamics (Levin and Nohel [11]). The Liapunov function here, having its root in the work of Levin [10], continues to play an important role in the investigation of Volterra equations. It is also well-known that under xq(x)>0 forx=0 withQ(x)→ ∞as|x| → ∞, condition (˜ii) with small modifications guarantees that the zero solution of the unperturbed equation

x(t)=

_t

−∞a(t−s)q

x(s)ds (3.51)

is asymptotically stable (Hale [9]).

Acknowledgment

References

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[10] J. J. Levin,The asymptotic behavior of the solution of a Volterra equation, Proceedings of the Amer-ican Mathematical Society14(1963), 534–541.

[11] J. J. Levin and J. A. Nohel,On a system of integro-differential equations occurring in reactor dy-namics, Journal of Mathematics and Mechanics9(1960), 347–368.

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Hang Gao: Department of Mathematics, Northeast Normal University, Changchun, Jilin 130024, China

E-mail address:[email protected]

Bo Zhang: Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC 28301-4298, USA